Your Private Key is a small piece of data that
identifies you, and you alone. Without your private key
no one can decrypt files or messages intended only for you
and no one can pretend to be you. That is, your private key is
needed to decrypt things intended for you and sign things that
purport to be from you. Never give your private key to
anyone for any reason.

Your private key is protected with the passphrase that you
entered when you generated it. Without the passphrase even
someone who steals the file containing your key can't use it.

Even so, don't let anyone have it. Guessing or finding your
passphrase is likely to many many times easier than guessing
or finding your private key. The difference is likely to be
days vs. centuries.

A key's Fingerprint uniquely identifies a particular
key. The fingerprint is usually used to verify that the key
you have is indeed the correct one. Anyone can generate a key
which claims to be from anyone else. But no one can generate a
key which claims to be someone else and has the same fingerprint
as their real key. (once again, this is backed up by math!)

Usually you would send your public key by email, then communicate
your key's fingerprint to the recipient by another means such as
a phone call, a letter, or in person. This ensures that no one
has intercepted the email with your key in it and replaced it
with their own. (That is known as a man-in-the-middle attack)